Is stock market overpriced? A benchmark approach

Figures & data

Table 1. Descriptive statistics of variables

Raw variables
	SPX	EARNING	PROFIT	IPI	SALES	BOND	REIT	GOLD	COMOD	OIL	GDPCAPITA	GDP	COUNTRY
Mean	516	26	485	59	952,838	351	4205	491	101	41	31,928	5892	0.36
Median	160	15	251	54	921,875	353	3794	371	82	30	29,508	3744	0.24
Maximum	2364	106	1890	107	1,426,160	676	10,344	1821	220	134	51,860	19,007	1.09
Minimum	17	2	34	15	478,951	101	1066	35	42	11	14,186	291	0.06
Std. dev.	597	28	542	29	258,020	167	2517	415	50	28	11,877	5709	0.28
Skewness	1.16	1.34	1.32	0.19	0.09	0.30	0.57	1.43	0.70	1.19	0.17	0.79	0.71
Kurtosis	3.16	3.58	3.45	1.69	1.79	1.92	2.24	4.18	2.08	3.31	1.65	2.27	2.19
Jarque–Bera	181	252	80	63	19	21	18	204	35	108	22	34	62
Probability	0	0	0	0	0	0	0	0	0	0	0	0	0
Ratios
Mean		18	0.99	6.34	0.0012	3.23	0.41	1.57	13	25	0.01	0.08	2,189,626
Median		17	0.99	2.95	0.0012	3.07	0.30	1.20	11	19	0.01	0.08	1,367,220
Maximum		126	2.60	22.80	0.0019	5.93	1.09	5.47	30	109	0.04	0.14	7,799,899
Minimum		7	0.35	1.06	0.0007	1.85	0.20	0.16	6	3	0.00	0.03	598,707
Observations	808	805	267	807	303	333	228	515	303	447	268	268	217

The first panel shows the descriptive statistics of all the variables. The second panel shows the descriptive statistics of the ratios of SPX to each variable

Earnings: The nominal earnings data are taken from Robert J. Shiller’s data base. Stock Market Data Used in “Irrational Exuberance” Princeton University Press, 2000, 2005, 2015, updated.

Profit: U.S. Bureau of Economic Analysis, Corporate business: Profits before tax (without IVA and CCAdj)

IPI: Board of Governors of the Federal Reserve System (US), Industrial Production Index

Sales: U.S. Bureau of the Census, Total Business Sales

Bonds: The proxy for bond performance is BofA Merrill Lynch US Corp a Total Return Index Value

REIT: Wilshire US Real Estate Investment Trust Total Market Index

Gold: ICE Benchmark Administration Limited (IBA), Gold Fixing Price 3:00 P.M. in London Bullion Market, based in U.S. Dollars

Comod: International Monetary Fund, Global Price Index of All Commodities

Oil: International Monetary Fund, Global price of WTI Crude

GDPCapita: U.S. Bureau of Economic Analysis, Real gross domestic product per capita

GDP: U.S. Bureau of Economic Analysis, Gross Domestic Product

Country: For each country “OECD, Total Share Prices for All Shares” series are converted to dollar for each month. The dollar value all share prices of 22 largest economies are averaged. These countries are Australia, Brazil, Canada, China, France, Germany, India, Indonesia, Italy, Japan, Korea, Mexico, Netherlands, New Zealand, Russia, South Africa, Saudi Arabia, Spain, Sweden, Switzerland, Turkey, and UK.

Figure 1. SPX versus P/E Ratio, Corporate Profit, Industrial Production and Total Business Sales

Figure 2. SPX versus Bond, REIT, Gold, and Commodity Index

Figure 3. SPX versus Oil, GDP, and Foreign Stocks

Figure 4. SPX versus Average Foreign Stocks, All Variables, Yardeni, and Shiller's CAPE

Table 2. Correlations among Spx and all other variables

BOND	0.89***	AUS	0.88***	NETHER	0.46***
COMOD	0.52***	BRAZ	0.32***	NZEL	0.83***
EARNING	0.94***	CANAD	0.89***	RUS	0.25***
GDP	0.96***	CHIN	0.53***	SAFRICA	0.79***
GDPCAPITA	0.90***	FRANCE	0.51***	SAUDI	0.34
GOLD	0.72***	GERMAN	0.75***	SPAIN	0.10
IPI	0.90***	INDI	0.85***	SWEDEN	0.10
OIL	0.58***	INDO	0.62***	SWISS	0.97***
PROFIT	0.93***	ITALY	−0.07	TUR	0.71***
REIT	0.85***	JAP	0.54***	UK	0.93***
SALES	0.85***	KORE	0.68***	COUNTRY	0.92***
		MEX	0.68***

***, **, * Significant at 1%, 5%, and 10% level, respectively. These are the pairwise correlations between SPX and all the variables listed in the table. The largest correlation is between Swiss Stock Market Index and SPX. GDP, earnings and profit are also highly correlated with SPX. Italy’s stock market index is negatively correlated with SPX. Commodities and oil have low correlation with SPX. These alternative assets could be valuable for diversification.

Table 3. Various regression models

Equation (3)		Equation (5)		Equation (6)
Variable	Coefficient	Variable	Coefficient	Variable	Coefficient
SWISS	950.1433	BOND	1.421976	C	−785.8227
SAFRICA	−2658.898	DUM	387.8779	FAIR	1.171751
OIL	4.281692	COMOD	−0.496559	DUM	850.3824
BOND	1.144757	C	−2255.804	GOLD	−0.081035
NETHER	555.0292	IPI	29.87364
SWEDEN	−507.9976
COMOD	−5.829598
CANAD	610.2987
R^2	0.972134		0.959721		0.690223
Adjusted R^2	0.971209		0.959160		0.686318
Included observations	219		292		242
Number of search regressor	33		_		_

All coefficients are significant at 1% precision level. The only exception is COMOD and GOLD. These two variables are not significant.

Table 4. Run counts and duration dependence test

Run Length	Positive runs	Negative runs
1	9	5
2	3	3
3	3	3
4	0	1
5	0	1
6	1	2
7	0	0
8	0	2
9	1	0
10	1	1
11	1	0
12	0	0
13	1	2
14	1	0
24	0	1
26	1	0
Log-logistic test
Alpha (α)	−0.61	−1.25***
Beta ($\beta$)	−0.62**	−0.2
LRT of $H_1:\beta =0$	6.21	0.56
p-value	0.01	0.45

Runs are consecutive positive and negative abnormal returns derived from the residuals from the regression model 4. ***, **, and * Significance of each coefficient at 1%, 5%, and 10% level, respectively. Alpha is the coefficient of the constant and beta is the coefficient of log run length. Log-likelihood test (LRT) tests the significance of beta. The restricted model assumes that the beta is zero. That is, there is no relationship between run length and probability of a market crash. The test is a chi-square test with one degree of freedom. The significantly negative beta coefficient for the positive runs indicates that there is an evidence for a rational bubble in the stock market. Negative coefficient suggests that as the run length increases, the probability of crash decreases.